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Properties

Label 8-5054e4-1.1-c1e4-0-2
Degree $8$
Conductor $6.524\times 10^{14}$
Sign $1$
Analytic cond. $2.65246\times 10^{6}$
Root an. cond. $6.35266$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s + 4·3-s + 10·4-s + 2·5-s + 16·6-s + 4·7-s + 20·8-s + 3·9-s + 8·10-s + 4·11-s + 40·12-s − 2·13-s + 16·14-s + 8·15-s + 35·16-s + 2·17-s + 12·18-s + 20·20-s + 16·21-s + 16·22-s + 4·23-s + 80·24-s − 10·25-s − 8·26-s − 10·27-s + 40·28-s + 12·29-s + ⋯
L(s)  = 1  + 2.82·2-s + 2.30·3-s + 5·4-s + 0.894·5-s + 6.53·6-s + 1.51·7-s + 7.07·8-s + 9-s + 2.52·10-s + 1.20·11-s + 11.5·12-s − 0.554·13-s + 4.27·14-s + 2.06·15-s + 35/4·16-s + 0.485·17-s + 2.82·18-s + 4.47·20-s + 3.49·21-s + 3.41·22-s + 0.834·23-s + 16.3·24-s − 2·25-s − 1.56·26-s − 1.92·27-s + 7.55·28-s + 2.22·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 7^{4} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 7^{4} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 7^{4} \cdot 19^{8}\)
Sign: $1$
Analytic conductor: \(2.65246\times 10^{6}\)
Root analytic conductor: \(6.35266\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 7^{4} \cdot 19^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(325.4334054\)
\(L(\frac12)\) \(\approx\) \(325.4334054\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_1$ \( ( 1 - T )^{4} \)
7$C_1$ \( ( 1 - T )^{4} \)
19 \( 1 \)
good3$C_4\times C_2$ \( 1 - 4 T + 13 T^{2} - 10 p T^{3} + 61 T^{4} - 10 p^{2} T^{5} + 13 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
5$((C_8 : C_2):C_2):C_2$ \( 1 - 2 T + 14 T^{2} - 28 T^{3} + 91 T^{4} - 28 p T^{5} + 14 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
11$((C_8 : C_2):C_2):C_2$ \( 1 - 4 T + 30 T^{2} - 96 T^{3} + 479 T^{4} - 96 p T^{5} + 30 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
13$((C_8 : C_2):C_2):C_2$ \( 1 + 2 T + 41 T^{2} + 86 T^{3} + 729 T^{4} + 86 p T^{5} + 41 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
17$((C_8 : C_2):C_2):C_2$ \( 1 - 2 T + 57 T^{2} - 70 T^{3} + 1341 T^{4} - 70 p T^{5} + 57 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
23$((C_8 : C_2):C_2):C_2$ \( 1 - 4 T + 78 T^{2} - 240 T^{3} + 2591 T^{4} - 240 p T^{5} + 78 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
29$((C_8 : C_2):C_2):C_2$ \( 1 - 12 T + 150 T^{2} - 1032 T^{3} + 6999 T^{4} - 1032 p T^{5} + 150 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
31$((C_8 : C_2):C_2):C_2$ \( 1 - 14 T + 150 T^{2} - 1096 T^{3} + 7139 T^{4} - 1096 p T^{5} + 150 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \)
37$((C_8 : C_2):C_2):C_2$ \( 1 - 24 T + 314 T^{2} - 2768 T^{3} + 18939 T^{4} - 2768 p T^{5} + 314 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \)
41$((C_8 : C_2):C_2):C_2$ \( 1 + 69 T^{2} - 310 T^{3} + 61 p T^{4} - 310 p T^{5} + 69 p^{2} T^{6} + p^{4} T^{8} \)
43$((C_8 : C_2):C_2):C_2$ \( 1 - 14 T + 203 T^{2} - 1740 T^{3} + 13581 T^{4} - 1740 p T^{5} + 203 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \)
47$((C_8 : C_2):C_2):C_2$ \( 1 + 2 T + 62 T^{2} - 320 T^{3} + 691 T^{4} - 320 p T^{5} + 62 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
53$((C_8 : C_2):C_2):C_2$ \( 1 + 122 T^{2} + 160 T^{3} + 8219 T^{4} + 160 p T^{5} + 122 p^{2} T^{6} + p^{4} T^{8} \)
59$((C_8 : C_2):C_2):C_2$ \( 1 - 2 T + 150 T^{2} - 552 T^{3} + 10859 T^{4} - 552 p T^{5} + 150 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
61$((C_8 : C_2):C_2):C_2$ \( 1 - 2 T + 118 T^{2} - 204 T^{3} + 10355 T^{4} - 204 p T^{5} + 118 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
67$((C_8 : C_2):C_2):C_2$ \( 1 - 22 T + 287 T^{2} - 2200 T^{3} + 17361 T^{4} - 2200 p T^{5} + 287 p^{2} T^{6} - 22 p^{3} T^{7} + p^{4} T^{8} \)
71$((C_8 : C_2):C_2):C_2$ \( 1 - 4 T + 120 T^{2} - 36 T^{3} + 6014 T^{4} - 36 p T^{5} + 120 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
73$((C_8 : C_2):C_2):C_2$ \( 1 - 10 T + 177 T^{2} - 1430 T^{3} + 16989 T^{4} - 1430 p T^{5} + 177 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
79$((C_8 : C_2):C_2):C_2$ \( 1 - 2 T + 75 T^{2} - 1132 T^{3} + 1589 T^{4} - 1132 p T^{5} + 75 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2^2:C_4$ \( 1 + 132 T^{2} + 17814 T^{4} + 132 p^{2} T^{6} + p^{4} T^{8} \)
89$((C_8 : C_2):C_2):C_2$ \( 1 - 10 T + 241 T^{2} - 2630 T^{3} + 28261 T^{4} - 2630 p T^{5} + 241 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
97$((C_8 : C_2):C_2):C_2$ \( 1 + 22 T + 322 T^{2} + 2860 T^{3} + 26911 T^{4} + 2860 p T^{5} + 322 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−5.85132130732452509057573921118, −5.41608685167298953714819761136, −5.38775464315394932456941883392, −5.20220385969077517562936652046, −4.93398945983642171303366542292, −4.71093634930739973230144983544, −4.65362223926869724288838949447, −4.42523548841550712885935667163, −4.25735846378586770874733942926, −3.85757018882641254973810120661, −3.83134084106193867614487197383, −3.77156651045790969730676136652, −3.54600716862151507285943611099, −2.98398284048836274358395053942, −2.92528775495107507666753693947, −2.83098161513961442544327904467, −2.63527313360761163849407069768, −2.36163631705371625125259544413, −2.32692829396212586946218037521, −2.05185018856517626124244202675, −2.03519887156618504649851682513, −1.36110442496875124777897286693, −1.15828199040227841278098924944, −0.907580472179705734363070532284, −0.822238766108433118248499429621, 0.822238766108433118248499429621, 0.907580472179705734363070532284, 1.15828199040227841278098924944, 1.36110442496875124777897286693, 2.03519887156618504649851682513, 2.05185018856517626124244202675, 2.32692829396212586946218037521, 2.36163631705371625125259544413, 2.63527313360761163849407069768, 2.83098161513961442544327904467, 2.92528775495107507666753693947, 2.98398284048836274358395053942, 3.54600716862151507285943611099, 3.77156651045790969730676136652, 3.83134084106193867614487197383, 3.85757018882641254973810120661, 4.25735846378586770874733942926, 4.42523548841550712885935667163, 4.65362223926869724288838949447, 4.71093634930739973230144983544, 4.93398945983642171303366542292, 5.20220385969077517562936652046, 5.38775464315394932456941883392, 5.41608685167298953714819761136, 5.85132130732452509057573921118

Graph of the $Z$-function along the critical line